Radiation View Factors

RADIATIVE VIEW FACTORS

View factor definition ................................................................................................................................... 2

View factor algebra ................................................................................................................................... 2

With spheres .................................................................................................................................................. 3

Patch to a sphere ....................................................................................................................................... 3

Frontal ................................................................................................................................................... 3

Level...................................................................................................................................................... 3

Tilted ..................................................................................................................................................... 3

Disc to frontal sphere ................................................................................................................................ 3

Cylinder to large sphere ............................................................................................................................ 4

Cylinder to its hemispherical closing cap ................................................................................................. 4

Sphere to sphere ........................................................................................................................................ 5

Small to very large ................................................................................................................................ 5

Equal spheres ........................................................................................................................................ 5

Concentric spheres ................................................................................................................................ 5

Hemispheres .......................................................................................................................................... 6

With cylinders ............................................................................................................................................... 6

Cylinder to large sphere ............................................................................................................................ 6

Cylinder to its hemispherical closing cap ................................................................................................. 6

Concentric very-long cylinders ................................................................................................................. 6

Concentric very-long cylinder to hemi-cylinder ....................................................................................... 7

Wire to parallel cylinder, infinite extent ................................................................................................... 7

Parallel very-long external cylinders ........................................................................................................ 7

Base to finite cylinder ............................................................................................................................... 7

Equal finite concentric cylinders ............................................................................................................... 8

With plates and discs..................................................................................................................................... 8

Parallel configurations .............................................................................................................................. 8

Equal square plates................................................................................................................................ 8

Unequal coaxial square plates ............................................................................................................... 9

Box inside concentric box ..................................................................................................................... 9

Equal rectangular plates ...................................................................................................................... 10

Equal discs .......................................................................................................................................... 10

Unequal discs ...................................................................................................................................... 11

Strip to strip ......................................................................................................................................... 11

Patch to infinite plate .......................................................................................................................... 11

Patch to disc ........................................................................................................................................ 11

Perpendicular configurations .................................................................................................................. 12

Square plate to rectangular plate ......................................................................................................... 12

Rectangular plate to equal rectangular plate ....................................................................................... 12

Rectangular plate to unequal rectangular plate ................................................................................... 12

Strip to strip ......................................................................................................................................... 13

Tilted configurations ............................................................................................................................... 13

Equal adjacent strips ........................................................................................................................... 13

Triangular prism .................................................................................................................................. 13

References ............................................................................................................................................... 13

VIEW FACTOR DEFINITION

The view factor F12 is the fraction of energy exiting an isothermal, opaque, and diffuse surface 1 (by

emission or reflection), that directly impinges on surface 2 (and is absorbed or reflected). Some view

factors having an analytical expression are compiled below.

View factors only depend on geometry, and can be computed from the general expression below.

Consider two infinitesimal surface patches, dA1 and dA2 (Fig. 1), in arbitrary position and orientation,

defined by their separation distance r12, and their respective tilting relative to the line of centres, 1 and

2, with 01/2 and 02/2 (i.e. seeing each other). The radiation power intercepted by surface dA2

coming directly from a diffuse surface dA1 is the product of its radiance L1=M1/, times its perpendicular

area dA1, times the solid angle subtended by dA2, d12; i.e.

d212=L1dA1d12=L1(dA1cos(1))dA2cos(2)/r12

2. Thence:

1 2

21 12 1 1 1 1 2 212

12 12 2

1 1 1 1 12

1 2 1 212 2 12 2 12 2

12 1 12

d d cos cos cos d cosdd d

d d

cos cos cos cos1d d d d

A A

LF

M M r

F F A Ar A r

Fig. 1. Geometry for view-factor definition.

When finite surfaces are involved, computing view factors is just a problem of mathematical integration

(not a trivial one, except in simple cases). Recall that the emitting surface (exiting, in general) must be

isothermal, opaque, and Lambertian (a perfect diffuser), and, to apply view-factor algebra, all surfaces

must be isothermal, opaque, and Lambertian.

View factor algebra

When considering all the surfaces under sight from a given one (enclosure theory), several general

relations can be established among the N2 possible view factors, what is known as view factor algebra:

Bounding. View factors are bounded to 0Fij≤1 by definition (the view factor Fij is the fraction

of energy exiting surface i, that impinges on surface j).

Closeness. Summing up all view factors from a given surface in an enclosure, including the

possible self-view factor for concave surfaces, 1ij

j

F , because the same amount of radiation

emitted by a surface must be absorbed.

Reciprocity. Noticing from the above equation that dAidFij=dAjdFji=(cosicosj/(rij2))dAidAj, it

is deduced that i ij j jiA F A F .

Distribution. When two target surfaces are considered at once, ,i j k ij ikF F F , based on area

additivity in the definition.

Composition. Based on reciprocity and distribution, when two source areas are considered

together, ,i j k i ik j jk i jF A F A F A A .

For an enclosure formed by N surfaces, there are N2 view factors (each surface with all the others and

itself). But only N(N1)/2 of them are independent, since another N(N1)/2 can be deduced from

reciprocity relations, and N more by closeness relations. For instance, for a 3-surface enclosure, we can

define 9 possible view factors, 3 of which must be found independently, another 3 can be obtained from

i ij j jiA F A F , and the remaining 3 by 1ij

j

F .

WITH SPHERES

Patch to a sphere

Frontal

Case View factor Plot

From a small planar plate

facing a sphere of radius

R, at a distance H from

centres, with hH/R.

12 2

1F

h

(e.g. for h=2, F12=1/4)

Level



level to a sphere of radius

R, at a distance H from

centres, with hH/R.

12 2

1 1arctan

xF

x h

with 2 1x h

(12 1

1 2 21

2hF h

)

(e.g. for h=2, F12=0.029)

Tilted



tilted to a sphere of

radius R, at a distance H

from centres, with

hH/R; the tilting angle

is between the normal

and the line of centres.

-if ||</2arcsin(1/h) (i.e. hcos>1),

12 2

cosF

h

-if not,

2

12 2

2

1cos arccos sin 1

sin 11arctan

F y x yh

y

x

with 2 1, cotx h y x

(e.g. for h=2 and =/4 (45º), F12=0.177)

Disc to frontal sphere


From a disc of radius R1

to a frontal sphere of

radius R2 at a distance H

between centres (it must

be H>R1), with hH/R1

and r2R2/R1.

2

12 2

2

12 1

11

F r

h

(e.g. for h=r2=1, F12=0.586)

From a sphere of radius

R1 to a frontal disc of

radius R2 at a distance H

between centres (it must

be H>R1, but does not

depend on R1), with

hH/R2.

12

2

1 11

2 11

F

h

(e.g. for R2=H and R1≤H, F12=0.146)

Cylinder to large sphere


From a small cylinder

(external lateral area

only), at an altitude

H=hR and tilted an angle

, to a large sphere of

radius R, is between

the cylinder axis and the

line of centres).

Coaxial (=0):

12

arcsin1

2 1

ssF

h

with 22

1

h hs

h

Perpendicular (=/2):

1

1

12 2 20

E d4

1

h x x xF

x

with elliptic integrals E(x).

Tilted cylinder:

1

arcsin21 2

12 2

0 0

sin 1 d dh zF

with

cos cos

sin sin cos

z

(e.g. for h=1 and any , F12=1/2)

Cylinder to its hemispherical closing cap


From a finite cylinder

(surface 1) of radius R

and height H, to its

hemispherical closing

cap (surface 2), with

r=R/H. Let surface 3 be

the base, and surface 4

the virtual base of the

hemisphere.

11 12

F

, 12 13 14

4F F F

214

Fr

,

22

1

2F ,

23

1

2 4F

r

,

312

Fr

,

32 12

Fr

,

34 12

Fr

with 24 1 1r

r

(e.g. for R=H, F11=0.38, F12=0.31,

F21=0.31, F22=0.50, F23=0.19,

F31=0.62, F32=0.38, F34=0.38)

Sphere to sphere

Small to very large


From a small sphere of

radius R1 to a much

larger sphere of radius R2

at a distance H between

centres (it must be H>R2,

but does not depend on

R1), with hH/R2.

12 2

1 11 1

2F

h

(e.g. for H=R2, F12=1/2)

Equal spheres



R to an equal sphere at a

distance H between

centres (it must be

H>2R), with hH/R.

12 2

1 11 1

2F

h

(e.g. for H=2R, F12=0.067)

Concentric spheres


Between concentric

spheres of radii R1 and

R2>R1, with rR1/R2<1.

F12=1

F21=r2

F22=1r2

(e.g. for r=1/2, F12=1, F21=1/4, F22=3/4)

Hemispheres


From a hemisphere of

radius R (surface 1) to its

base circle (surface 2).

F21=1

F12=A2F21/A1=1/2

F11=1F12=1/2

From a hemisphere of

radius R1 to a larger

concentric hemisphere of

radius R2>R1, with

RR2/R1>1. Let the

closing planar annulus be

surface 3.

12 14

F

, 13

4F

,

21 2

11

4F

R

,

22 2

1 11

2F

R

,

23 2 2

1 11 1

2 2 1F

R R

,

31 22F

R

,

32 21

2 1F

R

with

2 21 1 11 2 arcsin

2R R

R

(e.g. for R=2, F12=0.93, F21=0.23,

F13=0.07, F31=0.05, F32=0.95,

F23=0.36,F22=0.41)


R1 to a larger concentric

hemisphere of radius

R2>R1, with RR2/R1>1.

Let the enclosure be ‘3’.

F12=1/2, F13=1/2, F21=1/R2,

F23=1F21F22, 22 2

1 11

2F

R

with

2 21 1 11 2 arcsin

2R R

R

(e.g. for R=2, F12=1/2, F21=1/4, F13=1/2,

F23=0.34,F22=0.41)

WITH CYLINDERS

Cylinder to large sphere

See results under Cases with spheres.

Cylinder to its hemispherical closing cap

See results under Cases with spheres.

Concentric very-long cylinders


Between concentric

infinite cylinders of radii

R1 and R2>R1, with

rR1/R2<1.

F12=1

F21=r

F22=1r

(e.g. for r=1/2, F12=1, F21=1/2, F22=1/4)

Concentric very-long cylinder to hemi-cylinder


Between concentric

infinite cylinder of radius

R1 to concentric hemi-

cylinder of radius R2>R1,

with rR1/R2<1. Let the

enclosure be ‘3’.

F12=1/2, F21=r, F13=1/2,

F23=1F21F22,

2

22

21 1 arcsinF r r r

(e.g. for r=1/2, F12=1/2, F21=1/2,

F13=1/2, F23=0.22,F22=0.28)

Wire to parallel cylinder, infinite extent


From a small infinite

long cylinder to an

infinite long parallel

cylinder of radius R, with

a distance H between

axes, with hH/R.

12

1arcsin

hF

(e.g. for H=R, F12=1/2)

Parallel very-long external cylinders


From a cylinder of radius

R to an equal cylinder at

a distance H between

centres (it must be

H>2R), with hH/R.

2

12

24 2arcsin

2

h hhF

(e.g. for H=2R, F12=1/21/=0.18)

Base to finite cylinder


From base (1) to lateral

surface (2) in a cylinder

of radius R and height H,

with rR/H.

Let (3) be the opposite

base.

122

Fr

,

13 12

Fr

,

214

F

, 22 1

2F

,

234

F

with 24 1 1r

r

(e.g. for R=H, F12=0.62, F21=0.31,

F13=0.38, F22=0.38)

Equal finite concentric cylinders


Between finite concentric

cylinders of radius R1

and R2>R1 and height H,

with h=H/R1 and

R=R2/R1. Let the

enclosure be ‘3’. For the

inside of ‘1’, see

previous case.

2 412

1

11 arccos

2

f fF

f h

, 13 121F F ,

2

722

1 2 2 11 arctan

2

hfRF

R R h R

,

23 21 221F F F

with 2 2

1 1f h R , 2 2

2 1f h R ,

2 2

3 2 4f A R ,

2 14 3 2

1

1arccos arcsin

2

f ff f f

Rf R

,

2

5 2

41

Rf

h ,

2

6 2 2 2

21

4 4

hf

R h R

,

7 5 6 52

1arcsin arcsin 1 1

2f f f f

R

(e.g. for R2=2R1 and H=2R1, F12=0.64,

F21=0.34, F13=0.33, F23=0.43, F22=0.23)

WITH PLATES AND DISCS

Parallel configurations

Equal square plates


Between two identical

parallel square plates of

side L and separation H,

with w=W/H.

4

12 2 2

1ln 4

1 2

xF wy

w w

with 21x w and

arctan arctanw

y x wx

(e.g. for W=H, F12=0.1998)

Unequal coaxial square plates


From a square plate of

side W1 to a coaxial

square plate of side W2 at

separation H, with

w1=W1/H and w2=W2/H.

12 2

1

1ln

pF s t

w q

, with

22 2

1 2

2 2

2 1 2 1

2 2

2

2 2

,

arctan arctan

arctan arctan

4, 4

p w w

q x y

x w w y w w

x ys u x y

u u

x yt v x y

v v

u x v y

(e.g. for W1=W2=H, F12=0.1998)

Box inside concentric box


Between all faces in the

enclosure formed by the

internal side of a cube

box (faces 1-2-3-4-5-6),

and the external side of a

concentric cubic box

(faces (7-8-9-10-11-12)

of size ratio a1.

(A generic outer-box face

#1, and its corresponding

face #7 in the inner box,

have been chosen.)

From an external-box face:

11 12 13 14

2

15 16 17 18

19 1,10 1,11 1,12

0, , , ,

, , , ,

0, , ,

F F x F y F x

F x F x F za F r

F F r F r F r

From an internal-box face:

71 72 73 74

75 76 77 78

79 7,10 7,11 7,12

, 1 4, 0, 1 4,

1 4, 1 4, 0, 0,

0, 0, 0, 0

F z F z F F z

F z F z F F

F F F F

with z given by:

From face 1 to the others:

From face 7 to the others:

2

71 2

22

2

2

2

2

1ln

4

3 2 32

1

18 12 182

1

22arctan arctan

22arctan arctan

8 1 18, , 2

1 1

a pz F s t

a q

a ap

a

a aq

a

ws u w

u u

wt v w

v v

a au v w

a a

and:

2

2

1 4

0.2 1

1 4 4

r a z

y a

x y za r

(e.g. for a=0.5, F11=0, F12=0.16, F13=0.10,

F14=0.16, F15=0.16, F16=0.16, F17=0.20,

F18=0.01, F19=0, F1,10=0.01, F1,11=0.01,

F1,12=0.01), and (F71=0.79, F72=0.05, F73=0,

F74=0.05, F75=0.05, F76=0.05, F77=0, F78=0,

F79=0, F7,10=0, F7,11=0, F7,12=0).

Notice that a simple interpolation is proposed

for y≡F13 because no analytical solution has

been found.

Equal rectangular plates


Between parallel equal

rectangular plates of size

W1·W2 separated a

distance H, with x=W1/H

and y=W2/H.

2 2

1 112 2 2

1 1

1

1

1

1

1ln

1

2 arctan arctan

2 arctan arctan

x yF

xy x y

xx y x

y

yy x y

x

with 2

1 1x x and 2

1 1y y

(e.g. for x=y=1, F12=0.1998)

Equal discs



coaxial discs of radius R

and separation H, with

r=R/H.

2

12 2

1 4 11

2

rF

r

(e.g. for r=1, F12=0.382)

Unequal discs


From a disc of radius R1

to a coaxial parallel disc

of radius R2 at separation

H, with r1=R1/H and

r2=R2/H.

122

x yF

with 2 2 2

1 2 11 1x r r r and

2 2 2

2 14y x r r

(e.g. for r1=r2=1, F12=0.382)

Strip to strip



parallel strips of width W

and separation H, with

h=H/W.

2

12 1F h h

(e.g. for h=1, F12=0.414)

Patch to infinite plate


From a finite planar plate

at a distance H to an

infinite plane, tilted an

angle .

Front side: 12

1 cos

2F

Back side: 12

1 cos

2F

(e.g. for =/4 (45º),

F12,front=0.854, F12,back=0.146)

Patch to disc


From a patch to a parallel

and concentric disc of

radius R at distance H,

with h=H/R.

12 2

1

1F

h

(e.g. for h=1, F12=0.5)

Perpendicular configurations

Square plate to rectangular plate


From a square plate of

with W to an adjacent

rectangles at 90º, of

height H, with h=H/W.

2

12 1 2

1

1 1 1 1arctan arctan ln

4 4

hF h h h

h h

with 2

1 1h h and

4

12 2 22

hh

h h

(e.g. for h=→∞, F12=→1/4,

for h=1, F12=0.20004,

for h=1/2, F12=0.146)

Rectangular plate to equal rectangular plate


Between adjacent equal

rectangles at 90º, of

height H and width L,

with h=H/L.

12

1 2

1 1 12arctan 2 arctan

2

1ln

4 4

Fh h

h h

h

with 2

1 2 1h h and

22 1

2

1

11

h

hh

(e.g. for h=1, F12=0.20004)

Rectangular plate to unequal rectangular plate


From a horizontal

rectangle of W·L to

adjacent vertical

rectangle of H·L, with

h=H/L and w=W/L.

2 2

12

2 2

2 2

1 1 1arctan arctan

1arctan

1ln

4

w h

F h ww h w

h wh w

ab c

with 2 2

2 2

1 1

1

h wa

h w

,

2 2 2

2 2 2

1

1

w h wb

w h w

,

2 2 2

2 2 2

1

1

h h wc

h h w

(e.g. for h=w=1, F12=0.20004)

From non-adjacent

rectangles, the solution

can be found with view-

factor algebra as shown

here

2 2' 2'1 2 1 2 2' 1 2' 2 2' 1 2' 1

1 1

A AF F F F F

A A

2 2' 2'2 2' 1 1' 2 2' 1' 2' 1 1' 2' 1'

1 1

A AF F F F

A A

Strip to strip


Adjacent long strips at

90º, the first (1) of width

W and the second (2) of

width H, with h=H/W.

2

12

1 1

2

h hF

(e.g. 12

21 0.293

2H WF

)

Tilted configurations

Equal adjacent strips


Adjacent equal long

strips at an angle .

12 1 sin2

F

(e.g. 122

21 0.293

2F )

Triangular prism


Between two sides, 1 and

2, of an infinite long

triangular prism of sides

L1, L2 and L3 , with

h=L2/L1 and being the

angle between sides 1

and 2.

1 2 312

1

2

2

1 1 2 cos

2

L L LF

L

h h h

(e.g. for h=1 and =/2, F12=0.293)

References

Howell, J.R., “A catalog of radiation configuration factors”, McGraw-Hill, 1982.

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